Use the Newton-Raphson method to find the roots of the function
$f(x) = x^5 - 14.3x^4 + 76.15x^3 - 185.525x^2 + 202.3x - 89.625$, with an initial guess of $x_0 = 1$.
How can we prove the convergence of the iteration $x_{n+1} = x_n - f(x_n)/f'(x_n)$.I have tried several times, but it seems that I cannot find a suitable approach to prove it.
This is basically Darboux theorem.
As long as $$f(x_0) \times f''(x_0) >0$$ the method will converge to the solution with no overshoot. This is you case.
It is simple to find the intervals where $f''(x)$ is positive or negative. Then, select the $x_0$ accordingly.
Just for a stupid illustration, choose $x_0=123$ and run some $30$ iterations. No problem.
Now, choose $x_0=2$ and ...... let me know